I believe that they do. In fact, I believe that our dogs and cats (and maybe more than they) show us with each new day that they are going to Heaven.
Anything that possesses the capacity to love, has a soul.
And our pets love us unconditionally.
I cannot believe that God would create anything so, without them having an eternal destiny.
Dogs and cats have the same range of emotions humans have, anyone that has had a pet for a number of years knows this. Any creature with a capacity for love has a soul, I truly believe this. And this isn't hopium speaking either, it just makes sense in my mind. Our Father in Heaven is a magnificent creator and nothing is beyond possibility.